🎢 Halo Orbits & Invariant Manifolds

Invariant manifolds about Halo orbits and their applications in the Circular Restricted Three-body Problem.

• Joe Carpinelli

• December, $2020$

• ENAE$601$ Final Project

Presentation Mode: Toggle Presentation

📋 Project Overview

In the context of astrodynamics, manifolds are groups of trajectories that move toward or away from Lagrange points. To use invariant manifolds for interplanetary travel, several concepts must be developed and built on: lagrange points, periodic and quasi-periodic orbits within the Circular Restricted Three-body Problem, and manifolds about periodic orbits $\mathtt{\text{[1]}}$.

Outline

• Brief review of Lagrange points

• Periodic orbits (specifically, the subsection known as Halo orbits)

• Finding Halo orbits (both analytically, and numerically)

• Invariant manifolds about Halo orbits

Primary Reference

• Megan Rund's Masters Thesis at California Polytechnic State University $\mathtt{\text{[1]}}$

Lagrange Points

• Lagrange points are equilibrium points within the Circular Restricted Three-body Problem

Stability at Lagrange Points

• Like equilibrium points for all nonlinear systems, Lagrange points can be stable or unstable

• We can find the stability of an equilibrium point by analysing eigenvalues of the Jacobian of the state vector

In the $2$D case:

$$\left[\begin{array}{c}\dot{\zeta }\\ \dot{\beta }\\ \ddot{\zeta }\\ \ddot{\eta }\end{array}\right]=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ U_{xx}& U_{xy}& 0& 2\\ U_{yx}& U_{yy}& -2& 0\end{array}\right]\left[\begin{array}{c}\zeta \\ \eta \\ \dot{\zeta }\\ \dot{\eta }\end{array}\right]$$

$${\lambda }^4+(4-U_{xx}-U_{yy}){\lambda }^2+U_{xx}{U}_{yy}-U_{xy}^2=0$$

Stability at Lagrange Points: Examples

• Earth-Moon L4 is stable, and Earth-Moon L2 is unstable

)

Periodic Orbits

Libration Orbit Families

• Orbits about Lagrange points are known as Libration orbits

• Lyapunov orbits are two-dimensional Libration orbits

• Lissajous orbits are three-dimensional, semi-periodic Libration orbits

• Halo orbits are three-dimensional, theoretically periodic Libration orbits

Finding Halo Orbits

• The calculations required to solve for Halo orbits are numerically sensitive

• We covered the procedure for iteratively solving for Halo orbits numerically in Lecture 16; this requires an initial guess

• How do we find an initial guess for a Halo orbit?

😇 Analytical Halo Solution

• The Circular Restricted Three-body Dynamics can be derived using "Legendre poly- nomials", and there exists a third-order approximation (shown below) $\mathtt{\text{[5]}}$

• We can choose parameters to remove unstable (secular) terms from the expansion

Third Order CR3BP Expansion

Algorithm Inputs

• Nondimensional mass parameter $\mu$

• Nondimensional $Z-axis$ amplitude for the desired Halo orbit

• Lagrange point to orbit (L1 or L2)

Algorithm Outputs

• Position vector ${\overrightarrow{r}}_0$

• Velocity vector ${\overrightarrow{v}}_0$

• Estimated orbital period $T$

Analytical Halo Examples: Earth-Moon L2

NOT numerically propagated!

Analytical Halo Examples: Sun-Jupiter L1

NOT numerically propagated!

🧮 Numerical Halo Solution

• As discussed in Lecture 16, we can append the state transition matrix $\mathrm{\Phi }(t_0+t_i,t_0)$ to our state vector, and iteratively change initial conditions to numerically find a periodic orbit $\mathtt{\text{[1]}}$

1. $\mathrm{\Phi }=I,{\overrightarrow{x}}_0=\left[\begin{array}{cccccccccccc}{x}_0& 0& z_0& 0& {\dot{y}}_0& 0& {\mathrm{\Phi }}_1& {\mathrm{\Phi }}_2& {\mathrm{\Phi }}_3& {\mathrm{\Phi }}_4& {\mathrm{\Phi }}_5& {\mathrm{\Phi }}_6\end{array}\right]$

2. Propagate until $y=0$ again; $\dot{\mathrm{\Phi }}=F\mathrm{\Phi }$ where $F=\left[\begin{array}{cc}0& I_3\\ U_{XX}& 2\mathrm{\Omega }\end{array}\right]$, $U_{XX}$ is the matrix of second partial derivatives of potential $U$

3. Calculate $\left[\begin{array}{c}\delta x_0\\ \delta {\dot{y}}_0\end{array}\right]={\left(\left[\begin{array}{cc}{\mathrm{\Phi }}_{41}& {\mathrm{\Phi }}_{45}\\ {\mathrm{\Phi }}_{61}& {\mathrm{\Phi }}_{65}\end{array}\right]\frac{1}{\dot{y}}\left[\begin{array}{c}\ddot{x}\\ \ddot{z}\end{array}\right]\right)}^{-1}\left[\begin{array}{c}-\dot{x}\\ -\dot{z}\end{array}\right]$

4. Set $x_0\leftarrow x_0+\delta x_0$, ${\dot{y}}_0\leftarrow {\dot{y}}_0+\delta \dot{y}$ and jump back to Step 1 (until $\dot{x}$, $\dot{z}$ are both within some tolerance of zero)

🎢 Dynamics along Halo Orbits

Manifolds Exist

• Each point along a Halo orbit is connected with an unstable manifold, and a stable manifold

• The unstable manifold departs the Halo orbit, and the stable manifold arrives at the Halo orbit

• How can we calculate the perturbation required to shift the spacecraft onto the manifold?

Finding Manifolds

• We can use Eigenvectors of the Jacobian to calculate a state perturbation which will place the spacecraft onto a manifold at each point along the Halo orbit

1. Propagate the Halo orbit for one period $T$, including the state transition matrix $\mathrm{\Phi }(t_0+t_i,t_0)$

2. Let the final state transition matrix be $M$; $M=\mathrm{\Phi }(t_0+T,t_0)$

3. Calculate eigenvectors $V^S\leftrightarrow \text{minreal}(\text{eig}(M))$, and $V^U\leftrightarrow \text{maxreal}(\text{eig}(M))$

4. For each point $i$ along the Halo orbit...

   • $V_i^S=\mathrm{\Phi }(t_0+t_i,t_0)V^S$, and $V_i^U=\mathrm{\Phi }(t_0+t_i,t_0)V^U$

   • $X_i^S=X_i\pm ϵ\frac{V_i^S}{|V_i^S|}$, and $X_i^U=X_i\pm ϵ\frac{V_i^U}{|V_i^U|}$

Invariant Manifold Example

• Pulled from Rund's Thesis $\mathtt{\text{[1]}}$

• Note that stable manifolds need to be propagated backward in time from the perturbation along the Halo orbit, becuase they converge on the Halo orbit $\mathtt{\text{[1]}}$

🪐 Manifold-based Transfer Designs

• Design Overview summarized from $\mathtt{\text{[1]}}$

Design Overview

1. Find a desired Sun-Earth Halo orbit

2. Place the spacecraft within the stable manifold of this Halo orbit

3. Perturb the spacecraft from the Halo onto the unstable manifold to return towards Earth, and apply a maneuver to place the spacecraft on a Hyperbolic escape trajectory toward your destination planet

4. At destination planet, apply a maneuver to place spacecraft onto stable manifold of destination Halo orbit

🚀 Conclusions

Manifold Transfers

• Lagrange points, and periodic orbits about Lagrange points are surrounded by collections of trajectories called manifolds

• Manifolds can depart, or arrive at the Lagrange point / Libration orbit

• Stable manifolds can bring us to Halo orbits for free${}^{\ast }$ (plus the cost to place the spacecraft onto the manifold)

Lessons Learned

• Linear (Eigenvector) analysis works in Astrodynamics too

• Calculations for iterating on Halo orbits are extremely numerically sensitive

• Even with more complicated models (CR3BP), we need a patch-conic-like approach for interplanetary mission design

📚 References

$\mathtt{\text{[1]}}$ Rund, M. S., “Interplanetary Transfer Trajectories Using the Invariant Manifolds of Halo Orbits,” , 2018.

$\mathtt{\text{[2]}}$ Howell, K. C., “Three-dimensional, periodic,‘halo’orbits,” Celestial mechanics, Vol. 32, No. 1, 1984, pp. 53–71.

$\mathtt{\text{[3]}}$ Vallado, D. A., Fundamentals of astrodynamics and applications, Vol. 12, Springer Science & Business Media, 2001.

$\mathtt{\text{[4]}}$ Richardson, D., “Analytical construction of periodic orbits about the collinear points of the Sun-Earth system.” asdy, 1980, p. 127.

$\mathtt{\text{[5]}}$ Lara, M., Russell, R., and Villac, B., “Classification of the distant stability regions at Europa,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 2, 2007, pp. 409–418.

$\mathtt{\text{[6]}}$ Koon, W. S., Lo, M. W., Marsden, J. E., and Ross, S. D., “Dynamical systems, the three-body problem and space mission design,” Free online Copy: Marsden Books, 2008.

$\mathtt{\text{[7]}}$ Williams, J., Lee, D. E., Whitley, R. J., Bokelmann, K. A., Davis, D. C., and Berry, C. F., “Targeting cislunar near rectilinear halo orbits for human space exploration,” 2017.

$\mathtt{\text{[8]}}$ Zimovan-Spreen, E. M., Howell, K. C., and Davis, D. C., “Near rectilinear halo orbits and nearby higher-period dynamical structures: orbital stability and resonance properties,” Celestial Mechanics and Dynamical Astronomy, Vol. 132, No. 5, 2020, pp. 1–25.

$\mathtt{\text{[9]}}$ NASA, NASA’s Lunar Exploration Program Overview, 2020.

$\mathtt{\text{[10]}}$ Carpinelli, J., “UnitfulAstrodynamics.jl,” https://juliahub.com/ui/Packages/UnitfulAstrodynamics/uJGLZ/, 2020.

⌨️ Source Code

Package Dependencies

• The following packages were used – all are available in Julia's General package registry

Finding Lagrange Points

Unstable Lagrange Point (L2)

Stable Lagrange Point (L4)

Plot Analytical Halo Orbit

Northern Sun-Jupiter Halos

Southern Sun-Jupiter Halos

Northern Earth-Moon Halos

Southern Earth-Moon Halos

Numerically Produced Halo